Answer
$f'(x)=2x$
Work Step by Step
To take the derivative of a function using the limit process, plug into the equation $f'(x)=\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}$ and simplify:
$f'(x)=\lim\limits_{h \to 0}\frac{[(x+h)^2-5]-(x^2-5)}{h}$
$f'(x)=\lim\limits_{h \to 0}\frac{x^2+2xh+h^2-5-x^2+5}{h}$
$f'(x)=\lim\limits_{h \to 0}\frac{2xh+h^2}{h}$
$f'(x)=\lim\limits_{h \to 0}2x+h$
Once you can't simplify any further, plug $0$ in for $h$.
$f'(x)=2x+(0)=2x$
